36-401 Modern Regression HW #6 Solutionslarry/=stat401/HW6sol.pdf · 36-401 Modern Regression HW #6...

36-401 Modern Regression HW #6 SolutionsDUE: 10/27/2017 at 3PM

Problem 1 [32 points]

(a) (4 pts.)

5015

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Given : Chick

Time

Wei

ght

Figure 1: Weight Progression of 50 Chicks in the ChickWeight Data Set

1

(b) (9 pts.)

The results of a naive simple linear fit of Chick Weight on Time are shown in Figure 2. The right panel showsvery significant residual correlation. Therefore, the model does not fit well.

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esid

uals

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010

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Figure 2: Results of Naive Linear Regression of Chick 6 Weight on Time

Figure 3 displays the results of a degree four1 polynomial regression of Weight on Time. The lack of visiblecorrelation in the residuals suggests this is a much more suitabe fit.

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Time Time

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Figure 3: Results of Polynomial Regression of Chick 6 Weight on Time

1We chose this heuristically.

2

Table 1: Summary of Polynomial Regression for Chick 6

Estimate Std. Error t value Pr(>|t|)(Intercept) 42.7758402 3.2118729 13.318036 0.0000032Time -2.6030894 2.3499713 -1.107711 0.3045914I(Timeˆ2) 2.1242507 0.4898299 4.336711 0.0034098I(Timeˆ3) -0.1324975 0.0359324 -3.687415 0.0077831I(Timeˆ4) 0.0023642 0.0008494 2.783231 0.0271716

Not surprisingly, given the polynomial shape of the weight progression of Chick 6, our polynomial regressiondoes a much better job fitting the data than the naive linear regression. The most notable aspect of ourmodel parameters is that Time is no longer statistically significant given its higher order terms.

(c) (9 pts.)

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Time Time

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−120

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Figure 4: Results of Naive Linear Regression of Chick Weight on Time

As we saw with Chick 6, and as we could infer from Figure 1, the weight progression of individual chicks isoften more suitable to model with a polynomial than a simple linear regression. However, when the data areaggregated over all chicks, a simple linear model is actually not a poor choice. The model fit and residualsare shown in Figure 4. For the most part the line fits the data fairly well. However, some residual curvatureis apparent in the earlier stages of the chick lifetime. Therefore, adding a quadratic term will likely benefitthe model. Figure 5 shows that the extra term does indeed level out the residuals nicely.

3

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Figure 5: Results of Polynomial Regression of Chick Weight on Time

(d) (9 pts.)

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Wei

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Time

Diet 1Diet 2Diet 3Diet 4

Figure 6: Results of Polynomial Regression of Chick Weight on Time and Diet

4

Simply adding Diet as a predictor without interacting it with Time amounts to the growth curve for eachdiet group having the same shape but possibly different intercepts. As we can see in Figure 6, such a modelspecification is not suitable in this setting. Given the HW instructions, it is fine if you do this as long as youcomment on the ill-suitedness of the model.

Since a chick’s birthweight is independent of the diet it is thereafter fed, it is more sensible to force eachdiet group to have the same intercept, but possibly differing curvatures. The fits of this model are shown inFigure 7 (superposed) and Figure 8 (separated into respective Diet groups).

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Diet 1Diet 2Diet 3Diet 4

Figure 7: Polynomial Regression of Chick Weight with Timeˆ2 and Diet Interacted

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Diet = 1

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Time

Diet = 4

Figure 8: Polynomial Regression of Chick Weight with Timeˆ2 and Diet Interacted

The residuals of the model are plotted vs. Time in Figure 9 (superposed) and Figure 10 (separated intorespective Diet groups). The residuals look pretty healthy overall, but there is some clear correlation in Dietgroups 1 and 4. A box plot can also be used to do diet-wise diagnostics of the residuals (see Figure 11).However, the significant heteroskedasticity in all diet groups renders this plot next to useless. Instead we aremuch better off examining Figure 10. Notice we are only a few predictor interactions shy of performing fourseparate regressions any way.

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Time

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0 Diet 1Diet 2Diet 3Diet 4

Figure 9: Residuals of chosen model

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Figure 10: Residuals of chosen model

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Diet

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Figure 11: Diet-wise Distribution of Residuals

9


(a) (9 pts.)

There are about 100 ways to do this. Here is one.

Let

X0 =

1...1

,

i.e. the intercept predictor, andX =

[X0 X1 X2 X3

].

Now notice X0 = X1 +X2 +X3. That is, the columns of X are linearly dependent. Now,

Columns of X are LD =⇒ det(X) = 0.

And,det(XTX) = det(XT ) · det(X) = det(X) · det(X) = 02 = 0.

det(XTX) = 0 so XTX is not invertible.

(b) (9 pts.)

Y <- c(33,36,35,35,31,29,31,29,37,39,36,36)X0 <- rep(1,12)X1 <- c(rep(1,4),rep(0,8))X2 <- c(rep(0,4),rep(1,4),rep(0,4))X3 <- c(rep(0,8),rep(1,4))X <- cbind(X0,X1,X2) # leave out X3

model7 <- lm(Y ~ X - 1) # leave out default intercept since we have already included onetmp <- summary(model7)$coefficientsrownames(tmp) <- c("(Intercept)","France","Italy")kable(tmp, caption = "Income Regression Summary")

Table 2: Income Regression Summary

Estimate Std. Error t value Pr(>|t|)(Intercept) 37.00 0.6400955 57.803877 0.0000000France -2.25 0.9052317 -2.485552 0.0346742Italy -7.00 0.9052317 -7.732827 0.0000290

All coefficients in the regression are significant, signalling a statistically significant difference between themean incomes of France, Italy, and the USA.

10

(c) (9 pts.)

As we defined the model in part (b), we have

E[Income | France] = β0 + β1 = 34.75 (thousand)

E[Income | Italy] = β0 + β2 = 30.00 (thousand)

E[Income | USA] = β0 = 37.00 (thousand)

(d) (9 pts.)

Continuing with the same notation, the true mean income of France is given by β0 + β1. Assuming the noiseis normally distributed implies

β0 ∼ N(β0, Var(β0)

)and β1 ∼ N

(β1, Var(β1)

)and thus

β0 + β1 ∼ N(β0 + β1, Var(β0 + β1)

)N(β0 + β1, Var(β0) + Var(β1) + 2 · Cov(β0 + β1)

).

Therefore, a 95% confidence interval for β0 + β1 (mean income of France) is given by

(β0 + β1)± z0.025 ·√

Var(β0 + β1).

Var(β0 + β1) is not known so we replace it with its unbiased estimator and use the t-distribution. This yieldsthe 95% confidence interval

(β0 + β1)± tn−1(0.025) ·√Var(β0 + β1)

=⇒ (β0 + β1)± t3(0.025) ·√

Var(β0) + Var(β1) + 2 · Cov(β0, β1)=⇒ 34.75± 2.037,

where we have gotten the estimated errors from the vcov function.

11


(a) [5 pts.]

age

100 200 0.0 0.4 0.8 0.0 0.4 0.8 0 2 4 6

1525

3545

100

200

lwt

race

1.0

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smoke

ptl

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ht

ui

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02

46

ftv

15 25 35 45 1.0 2.0 3.0 0.0 1.5 3.0 0.0 0.4 0.8 1000 4000

1000

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bwt

12

(b) [9 pts.]

Table 3: Summary of Birth Weight Regression

Estimate Std. Error t value Pr(>|t|)(Intercept) 2841.040247 328.894986 8.6381379 0.0000000age -2.955912 9.604270 -0.3077706 0.7586271lwt 4.380344 1.834985 2.3871276 0.0180591factor(race)2 -442.329304 148.926061 -2.9701269 0.0033999factor(race)3 -282.141331 120.748135 -2.3366103 0.0206050smoke -283.589210 111.748150 -2.5377531 0.0120397factor(ptl)1 -357.092406 152.324861 -2.3442818 0.0201992factor(ptl)2 -68.778992 297.501669 -0.2311886 0.8174415factor(ptl)3 1260.027892 661.664516 1.9043305 0.0585268ht -553.036031 202.192322 -2.7351980 0.0068840ui -522.713818 138.164206 -3.7832796 0.0002130factor(ftv)1 142.479511 122.824933 1.1600211 0.2476386factor(ftv)2 -1.868773 138.659587 -0.0134774 0.9892624factor(ftv)3 -319.779655 253.984296 -1.2590529 0.2097074factor(ftv)4 244.941583 336.965702 0.7269036 0.4682675factor(ftv)6 15.369792 700.858349 0.0219300 0.9825291

(c) [9 pts.]

Overall, the residuals look quite good plotted against the fitted values and the numerical predictors. Case226 (a 45 year old) bears substantial leverage on the fit since all other cases are much younger. Furthermore,this data point also yields a very large outlier. In practice, we would therefore discard this observation whenbuilding a predictive model. In regard to the categorical predictors, there is some interclass heteroskedasticity;however, the varied sample sizes can also make is challenging to deduce this strictly from these box plots.That is, categorical levels containing a large number of observations can give the appearance of a widerdistribution while levels with small sample sizes natural appear to have low variance. To normalize thesecomparisons with respect to sample size we could, for example, perform an F -test for equality of variances or(more sophisticated) a Kolmorogov-Smirnov test for equality of distributions.

13

Fitted values

Res

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lm(bwt ~ age + lwt + factor(race) + smoke + factor(ptl) + ht + ui + factor( ...

10

226

16

1800 2200 2600 3000 3400 3800

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−180

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Age

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80 100 120 140 160 180 200 220 240

−180

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0030

090

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1 2 3

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Race

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Smoke

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Number of previous premature labours

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0 1

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History of hypertension

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Presence of uterine irritability

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0 1 2 3 4 6

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Number of physician visits during the first trimester.

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(d) [9 pts.]

Table 3 (part b) suggests we may have included more predictors than are useful for predicting Birth Weight.A more predictive model would likely reduce the dimension of our covariates. We could, for example, proceedby performing a sequence of partial F -tests (equivalently stepwise regression; 36-402) or perhaps implementinga LASSO regression (also 36-402)2 Given the full set of predictors, the multiple linear regression suggeststhat a baby’s birth weight is significantly associated with the mother’s weight, race, smoking status, historyof hypertension, and presence of uterine iritability (and possibly the number of previous premature labours).

2Gee, I bet you guys can’t wait until 36-402.

18

Appendix

addTrans <- function(color,trans){

# This function adds transparancy to a color.# Define transparancy with an integer between 0 and 255# 0 being fully transparant and 255 being fully visable# Works with either color and trans a vector of equal length,# or one of the two of length 1.

if (length(color)!=length(trans)&!any(c(length(color),length(trans))==1)){stop("Vector lengths not correct")

}if (length(color)==1 & length(trans)>1) color <- rep(color,length(trans))if (length(trans)==1 & length(color)>1) trans <- rep(trans,length(color))

num2hex <- function(x){

hex <- unlist(strsplit("0123456789ABCDEF",split=""))return(paste(hex[(x-x%%16)/16+1],hex[x%%16+1],sep=""))

}rgb <- rbind(col2rgb(color),trans)res <- paste("#",apply(apply(rgb,2,num2hex),2,paste,collapse=""),sep="")return(res)

}


data(ChickWeight)names(ChickWeight)attach(ChickWeight)

(a) (4 pts.)

coplot(weight ~ Time | Chick, data = ChickWeight,type = "b", show.given = FALSE,xlab = "", ylab = "", main = "")

mtext(side = 1, text = "Time", cex = 2, line = 3.75)mtext(side = 2, text = "Weight", cex = 2, line = 2.25)

(b) (9 pts.)

Chick6 <- ChickWeight[ChickWeight$Chick == 6,]

par(mfrow=c(1,2))with(Chick6, plot(Time, weight, col = NA, pch = 19, cex = 1.25,

axes = FALSE, xlab="",ylab=""))axis(side = 1, at = c(0,5,10,15,20), as.character(c(0,5,10,15,20)),

font = 5)

19

axis(side = 2, at = seq(0,300,25), as.character(seq(0,300,25)),font = 5)

abline(v = c(0,5,10,15,20), h = seq(50,300,25), col = "gray70",lty = 2)

mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)with(Chick6, points(Time, weight, col = addTrans("orange",120),

cex = 1.25, pch = 19))with(Chick6, points(Time, weight, col = "orange", pch = 1,

cex = 1.25))model1 <- lm(weight ~ Time, data = Chick6)abline(model1, col = "seagreen", lwd = 2)

plot(Chick6$Time, residuals(model1), col = NA, axes = FALSE,xlab= "Time", ylab = "Residuals", font.lab = 3)

axis(side = 1, at = c(0,5,10,15,20), as.character(c(0,5,10,15,20)),font = 5)

axis(side = 2, at = seq(-70,30,10), labels = as.character(seq(-70,30,10)),font = 5)

abline(h = seq(-70,30,10), v = c(0,5,10,15,20), col = "gray70",lty = 2)

abline(0,0, lty = 2, col = "gray45")points(Chick6$Time, residuals(model1), col = addTrans("orange",120),

pch = 19, cex = 1.25)points(Chick6$Time, residuals(model1), col = "orange", cex = 1.25)panel.smooth(Chick6$Time, residuals(model1), col = NA,cex = 0.5,

col.smooth = "seagreen", span = 0.5, iter = 3)

par(mfrow=c(1,2))with(Chick6, plot(Time, weight, col = NA, pch = 19, cex = 1.25,

axes = FALSE, xlab="",ylab=""))axis(side = 1, at = c(0,5,10,15,20), as.character(c(0,5,10,15,20)),

font = 5)axis(side = 2, at = seq(0,300,25), as.character(seq(0,300,25)),

font = 5)abline(v = c(0,5,10,15,20), h = seq(50,300,25), col = "gray70",

lty = 2)mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)with(Chick6, points(Time, weight, col = addTrans("orange",120),

cex = 1.25, pch = 19))with(Chick6, points(Time, weight, col = "orange", pch = 1, cex = 1.25))model2 <- lm(weight ~ Time + I(Time ^ 2) + I(Time ^ 3) + I(Time ^ 4),

data = Chick6)lines(Chick6$Time, fitted(model2), col = "seagreen", lwd = 2)

plot(Chick6$Time, residuals(model2), col = NA, axes = FALSE, xlab= "Time",ylab = "Residuals", font.lab = 3)

axis(side = 1, at = c(0,5,10,15,20), as.character(c(0,5,10,15,20)),font = 5)


abline(h = seq(-70,30,10), v = c(0,5,10,15,20), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")

20

points(Chick6$Time, residuals(model2), col = addTrans("orange",120),pch = 19, cex = 1.25)

points(Chick6$Time, residuals(model2), col = "orange", cex = 1.25)panel.smooth(Chick6$Time, residuals(model2), col = NA,cex = 0.5,


library(knitr)kable(summary(model2)$coefficients, caption = "Summary of Polynomial Regression for Chick 6")

(c) (9 pts.)

par(mfrow=c(1,2))with(ChickWeight, plot(Time, weight, col = NA, pch = 19, cex = 0.8,

axes = FALSE, xlab="",ylab=""))axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)),

font = 5)axis(side = 2, at = seq(0,350,50), as.character(seq(0,350,50)),

font = 5)abline(v = c(0,5,10,15,20,25), h = seq(50,350,50), col = "gray70",

lty = 2)mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)with(ChickWeight, points(Time, weight, col = addTrans("orange",120),

cex = 0.8, pch = 19))with(ChickWeight, points(Time, weight, col = "orange", pch = 1,

cex = 0.8))model3 <- lm(weight ~ Time, data = ChickWeight)abline(model3, col = "seagreen", lwd = 2)

plot(ChickWeight$Time, residuals(model3), col = NA, axes = FALSE,xlab= "Time", ylab = "Residuals", font.lab = 3)

axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)),font = 5)


abline(h = seq(-200,200,40), v = c(0,5,10,15,20,25), col = "gray70",lty = 2)

abline(0,0, lty = 2, col = "gray45")points(ChickWeight$Time, residuals(model3), col = addTrans("orange",120),

pch = 19, cex = 0.8)points(ChickWeight$Time, residuals(model3), col = "orange", cex = 0.8)panel.smooth(ChickWeight$Time, residuals(model3), col = NA,cex = 0.5,


par(mfrow=c(1,2))with(ChickWeight, plot(Time, weight, col = NA, pch = 19, cex = 0.8,

axes = FALSE, xlab="",ylab=""))axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)),

font = 5)axis(side = 2, at = seq(0,350,50), as.character(seq(0,350,50)), font = 5)abline(v = c(0,5,10,15,20,25), h = seq(50,350,50), col = "gray70", lty = 2)mtext(side = 2, text = "Weight", font = 3, line = 3)

21

mtext(side = 1, text = "Time", font = 3, line = 3)with(ChickWeight, points(Time, weight, col = addTrans("orange",120), cex = 0.8, pch = 19))with(ChickWeight, points(Time, weight, col = "orange", pch = 1, cex = 0.8))model4 <- lm(weight ~ Time + I(Time ^ 2), data = ChickWeight)lines(seq(0,max(ChickWeight$Time),length = 50),

predict(model4, newdata=data.frame(Time = seq(0,max(ChickWeight$Time),length = 50))),col = "seagreen", lwd = 2)

plot(ChickWeight$Time, residuals(model4), col = NA, axes = FALSE, xlab= "Time",ylab = "Residuals", font.lab = 3)

axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)), font = 5)axis(side = 2, at = seq(-200,200,40), labels = as.character(seq(-200,200,40)),

font = 5)abline(h = seq(-200,200,40), v = c(0,5,10,15,20,25), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")points(ChickWeight$Time, residuals(model4), col = addTrans("orange",120),

pch = 19, cex = 0.8)points(ChickWeight$Time, residuals(model4), col = "orange", cex = 0.8)panel.smooth(ChickWeight$Time, residuals(model4), col = NA,cex = 0.5,


(d) (9 pts.)

with(ChickWeight, plot(Time, weight, col = NA, pch = 19, cex = 0.8,axes = FALSE, xlab="",ylab=""))

axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)), font = 5)axis(side = 2, at = seq(0,350,50), as.character(seq(0,350,50)), font = 5)abline(v = c(0,5,10,15,20,25), h = seq(50,350,50), col = "gray70", lty = 2)mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)with(ChickWeight, points(Time, weight, col = addTrans("orange",120),

cex = 0.8, pch = 19))with(ChickWeight, points(Time, weight, col = "orange", pch = 1, cex = 0.8))model5 <- lm(weight ~ Time + I(Time ^ 2) + factor(Diet), data = ChickWeight)lines(seq(0,max(ChickWeight$Time),length = 50),

predict(model5, newdata=data.frame(Time = seq(0,max(ChickWeight$Time),length = 50),Diet = 1)),

col = "seagreen", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "red", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "blue", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "purple", lwd = 2)legend(x = "topleft", legend = c("Diet 1", "Diet 2", "Diet 3", "Diet 4"),

col = c("seagreen","red","blue","purple"), lwd = 2)

22

with(ChickWeight, plot(Time, weight, col = NA, pch = 19, cex = 0.8, axes = FALSE,xlab="",ylab=""))

axis(side = 1, at = c(0,5,10,15,20,25), as.character(c(0,5,10,15,20,25)), font = 5)axis(side = 2, at = seq(0,350,50), as.character(seq(0,350,50)), font = 5)abline(v = c(0,5,10,15,20,25), h = seq(50,350,50), col = "gray70", lty = 2)mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)with(ChickWeight, points(Time, weight, col = addTrans("orange",120), cex = 0.8,

pch = 19))with(ChickWeight, points(Time, weight, col = "orange", pch = 1, cex = 0.8))model6 <- lm(weight ~ Time + I(Time ^ 2) : factor(Diet), data = ChickWeight)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "seagreen", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "red", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "blue", lwd = 2)lines(seq(0,max(ChickWeight$Time),length = 50),


col = "purple", lwd = 2)legend(x = "topleft", legend = c("Diet 1", "Diet 2", "Diet 3", "Diet 4"),

col = c("seagreen","red","blue","purple"), lwd = 2, bg = "white")

par(mfrow=c(2,2))cols <- c("seagreen","red","blue","purple")for (itr in 1:4){

with(ChickWeight, plot(Time, weight, col = NA, pch = 19, cex = 0.8,axes = FALSE, xlab="",ylab=""))


axis(side = 2, at = seq(0,350,50), as.character(seq(0,350,50)),font = 5)

abline(v = c(0,5,10,15,20,25), h = seq(50,350,50), col = "gray70",lty = 2)

mtext(side = 2, text = "Weight", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)mtext(side = 3, text = paste0("Diet = ",itr), font = 3)with(ChickWeight[which(ChickWeight$Diet==itr),], points(Time, weight,

col = addTrans("orange",120),cex = 0.8, pch = 19))

with(ChickWeight[which(ChickWeight$Diet==itr),], points(Time, weight,col = "orange",pch = 1, cex = 0.8))

model6 <- lm(weight ~ Time + I(Time ^ 2) : factor(Diet),data = ChickWeight)

lines(seq(0,max(ChickWeight$Time),length = 50),predict(model6, newdata=data.frame(Time = seq(0,max(ChickWeight$Time),

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length = 50), Diet = itr)),col = cols[itr], lwd = 2)

}

plot(ChickWeight$Time, residuals(model6), col = NA, axes = FALSE,xlab= "Time", ylab = "Residuals", font.lab = 3)



abline(h = seq(-200,200,40), v = c(0,5,10,15,20,25), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")points(ChickWeight$Time, residuals(model6),

col = addTrans(c("seagreen","red","blue","purple")[ChickWeight$Diet],120),pch = 19, cex = 0.8)

points(ChickWeight$Time, residuals(model6),col = c("seagreen","red","blue","purple")[ChickWeight$Diet], cex = 0.8)

legend(x = "topleft", legend = c("Diet 1", "Diet 2", "Diet 3", "Diet 4"),col = addTrans(c("seagreen","red","blue","purple"),120), pch = 19, bg = "white")

par(mfrow=c(2,2))cols <- c("seagreen","red","blue","purple")for (itr in 1:4){

with(ChickWeight, plot(Time, residuals(model6), col = NA,pch = 19, cex = 0.8, axes = FALSE, xlab="",ylab=""))



abline(h = seq(-200,200,40), v = c(0,5,10,15,20,25), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")mtext(side = 2, text = "Residuals", font = 3, line = 3)mtext(side = 1, text = "Time", font = 3, line = 3)mtext(side = 3, text = paste0("Diet = ",itr), font = 3)with(ChickWeight, points(Time[which(ChickWeight$Diet==itr)],

residuals(model6)[which(ChickWeight$Diet==itr)],col = addTrans(c("seagreen","red","blue","purple"),120)[itr],cex = 0.8, pch = 19))

with(ChickWeight, points(Time[which(ChickWeight$Diet==itr)],residuals(model6)[which(ChickWeight$Diet==itr)],col = c("seagreen","red","blue","purple")[itr], pch = 1,cex = 0.8))

}

boxplot(residuals(model6) ~ ChickWeight$Diet,col = addTrans(c("seagreen","red","blue","purple"),120),border = c("seagreen","red","blue","purple"),xlab = "Diet",ylab = "Residuals", pch = 19)

abline(0,0, lty = 2, col = "gray45")

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library(MASS)

(a) [5 pts.]

pairs(birthwt[,2:10])

(b) [9 pts.]

model8 <- lm(bwt ~ age + lwt + factor(race) + smoke + factor(ptl) + ht+ ui + factor(ftv), data = birthwt)

kable(summary(model8)$coefficients,caption = "Summary of Birth Weight Regression")

(c) [9 pts.]

plot(model8, which = 1, col = NA, pch = 19, axes = FALSE,add.smooth = FALSE, caption = "")

abline(h = seq(-2100,1500,300), col = "gray75", lty = 2)abline(v = seq(1800,3800,200), col = "gray80", lty = 2)abline(0,0, lty = 2, col = "gray45")axis(side = 1, at = seq(1800,3800,200),

as.character(seq(1800,3800,200)), font = 5)axis(side = 2, at = seq(-2100,1500,300),

labels = as.character(seq(-2100,1500,300)), font = 5)points(fitted(model8), residuals(model8),

col = addTrans("orange",120), pch = 19)points(fitted(model8), residuals(model8), col = "orange")panel.smooth(fitted(model8), residuals(model8), col = "orange",

cex = 1, col.smooth = "seagreen", span = 2/3, iter = 3)

plot(birthwt$age, residuals(model8), col = NA, axes = FALSE, xlab= "Age",ylab = "Residuals", font.lab = 3)

axis(side = 1, at = seq(10,50,5), as.character(seq(10,50,5)), font = 5)axis(side = 2, at = seq(-1800,1500,300),

labels = as.character(seq(-1800,1500,300)), font = 5)abline(h = seq(-1800,1500,300), v = seq(10,50,5), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")points(birthwt$age, residuals(model8),

col = addTrans("orange",120), pch = 19, cex = 0.8)points(birthwt$age, residuals(model8), col = "orange", cex = 0.8)panel.smooth(birthwt$age, residuals(model8), col = NA,cex = 0.5,

col.smooth = "seagreen", span = 2/3, iter = 3)

plot(birthwt$lwt, residuals(model8), col = NA, axes = FALSE,xlab= "Age", ylab = "Residuals", font.lab = 3)

axis(side = 1, at = seq(80,250,20), as.character(seq(80,250,20)), font = 5)axis(side = 2, at = seq(-1800,1500,300),

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labels = as.character(seq(-1800,1500,300)), font = 5)abline(h = seq(-1800,1500,300), v = seq(80,250,20), col = "gray70", lty = 2)abline(0,0, lty = 2, col = "gray45")points(birthwt$lwt, residuals(model8), col = addTrans("orange",120),

pch = 19, cex = 0.8)points(birthwt$lwt, residuals(model8), col = "orange", cex = 0.8)panel.smooth(birthwt$lwt, residuals(model8), col = NA,cex = 0.5,

col.smooth = "seagreen", span = 2/3, iter = 3)

boxplot(residuals(model8) ~ birthwt$race, col = addTrans(c("seagreen","red","blue"),120),border = c("seagreen","red","blue"), xlab = "Race",ylab = "Residuals", pch = 19)


boxplot(residuals(model8) ~ birthwt$smoke, col = addTrans(c("seagreen","orange"),120),border = c("seagreen","orange"), xlab = "Smoke",ylab = "Residuals", pch = 19)


boxplot(residuals(model8) ~ birthwt$ptl,col = addTrans(c("seagreen","orange","red","blue"),120),border = c("seagreen","orange","red","blue"),xlab = "Number of previous premature labours",ylab = "Residuals", pch = 19)


boxplot(residuals(model8) ~ birthwt$ht, col = addTrans(c("seagreen","orange"),120),border = c("seagreen","orange"), xlab = "History of hypertension",ylab = "Residuals", pch = 19)


boxplot(residuals(model8) ~ birthwt$ui, col = addTrans(c("seagreen","orange"),120),border = c("seagreen","orange"), xlab = "Presence of uterine irritability",ylab = "Residuals", pch = 19)


boxplot(residuals(model8) ~ birthwt$ftv,col = addTrans(c("seagreen","orange","red","blue","purple","yellow2","pink"),120),border = c("seagreen","orange","red","blue","purple","yellow2","pink"),xlab = "Number of physician visits during the first trimester.",ylab = "Residuals", pch = 19)


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36-401 Modern Regression HW #6 Solutionslarry/=stat401/HW6sol.pdf · 36-401 Modern Regression HW #6...

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